sinusoidal vibration measurements in a piezoelectric xy

Transcription

ABCM Symposium Series in Mechatronics - Vol. 6
Copyright © 2014 by ABCM
Part I - International Congress
Section II - Sensor, Vision, Image & Embedded Systems
SINUSOIDAL VIBRATION MEASUREMENTS IN A PIEZOELECTRIC XY
NANOPOSITIONER BY USING A NEW INTERFEROMETRIC METHOD
Andryos da Silva Lemes
José Henrique Galeti
Ricardo Tokio Higuti
Cláudio Kitano
Faculdade de Engenharia de Ilha Solteira, UNESP - Universidade Estadual Paulista, Department of Electrical Engineering,
Av. Brasil, 56, 15385-000 – Ilha Solteira, SP, Brazil
[email protected], [email protected], [email protected], [email protected]
Ronny Calixto Carbonari
Universidade Federal do ABC, Centro de Engenharia, Modelagem e Ciências Sociais Aplicadas, Rua Santa Adélia, 166,
09210-170 - Santo André, SP – Brazil
[email protected]
Emílio Carlos Nelli Silva
Department of Mechatronics and Mechanical Systems Engineering - Escola Politécnica da Universidade de São Paulo – EPUSP,
Av. Prof. Mello Moraes, 2231, 05508-900 – São Paulo, SP, Brazil
[email protected]
Abstract. Nanopositioning systems are fundamental in scanning probe microscopy and in microelectronics, where
elements with nanometer dimensions are manipulated. These systems require positioning stages with nanoscale
resolution for assembly, testing and characterization. It is recognized that piezoelectric actuators are ubiquitous tools
in a wide variety of applications, particularly those involving nanomanipulation. A multi-actuated piezoelectric
actuator is a device with small dimensions that can be driven by reduced voltages and can produce nanoscale
displacements. These devices consist in a multi-flexible structure actuated by two or more piezoceramics. Such
structure acts as a mechanical transformer by amplifying and changing the direction of the piezoceramic
displacements. In this work, a XY nanopositioner, designed by using the topology optimization technique, is tested.
Measurements of nanodisplacements require calibration by comparison with an absolute standard, and this is
performed by using interferometers. A low cost homodyne Michelson interferometer is used to detect the
nanovibrations of XY actuators, using a method based on the spectral analysis of the photodetected signal. This new
method is self-consistent, is immune to fading, does not demand Bessel functions algebraic sign correction algorithms,
does not present singularities when the static phase shift between the interferometer arms is equal to an integer
multiple of /2 rad, and has a wide dynamic range (from 0.18 to 130 rad). The XY nanopositioner characterization was
based on the determination of the actuator length-to-voltage sensitivities (in nm/V), displacement frequency response
and coupling rate between the generated and coupled displacements.
Keywords: nanopositioner XY, nanometric displacement, laser interferometry, PZT
1. INTRODUCTION
Nowadays there is a substantial interest in multi-actuated piezoelectric actuators (MAPFAs), which are solid state
mechanisms where the action is given by the structural flexibility and not by the presence of joints and pins (Carbonari
et al., 2005a). They are assembled in a single part and can also amplify and change the direction of piezoceramic
displacements. New MAPFAs models have been developed using the topology optimization technique, which is a
method of computational design that combines an optimization algorithm and finite element method (FEM) to obtain
the optimal topology of the mechanical parts.
The design of a MAPFA must take into account the displacement amplification provided by the flexible structure
and the force produced in the desired displacement direction. These parameters can be optimized if the structure is
designed to provide flexibility and stiffness. Numerical analysis with ANSYS software is used to model the devices,
taking into consideration all structure components: piezoceramics, adhesives, excitation type, clamped points, etc. The
goal of topology optimization is to determine, through the addition and removal of material, a continuous solid structure
that meets the specified objective function (Carbonari et al., 2007).
One potential application of MAPFAs is in the area of nanotechnology, which deals with objects whose size is
below 100 nm and thus below atomic scale dimensions (Zhelkobaev et al., 2007). The micro/nano manufacturing is an
emerging industry with significant market potential, but the available measurement tools are still complicated. Scanning
probe microscopy (SPM) has become the main technique for characterization, nanomechanical testing and
nanoassembly, and atomic force microscopy (AFM) is the most frequently used SPM. These systems require
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positioning mechanisms with nanoscale resolution and, consequently, fundamental issues in nanotechnology and future
advances in the field of SPM depend on stages of nano positioning with high resolution and high bandwidth (Devasia et
al., 2007).
It is well known that accuracy and resolution positioning of manipulators depend mainly on their topological and
mechanical structures, which should preferably be flexure-based mechanisms (Tian et al., 2009). Thus, over the years,
various arrangements have been proposed for nanometric positioners as, for example, the multi-actuated microtools
(Carbonari et al., 2007; Carbonari et al., 2005a) (as the MPFAs in this paper), the multi-actuated functionally graded
piezoelectric micro-tools (Carbonari et al., 2009), the flexure based hexapod nanopositioner (Shi et al., 2013), the
KoalaDrive nanopositioner (Cherepanov et al., 2012), the flexure-based five bar mechanism (Tian et al., 2009) and
many others (Devasia et al., 2007).
This suggests that the measurement of very small displacements with high accuracies is the foundation for a variety
of applications in nanotechnology, such as, for calibration of STM and AFM, holographic scales and many other
actuator types (Schibli et al., 2006). The interferometers are often used in nanopositioning devices and
nanomeasurements because use frequency stabilized lasers and the measurement is traceable to the meter definition and
national and international standards (Hausotte et al., 2012).
At the moment, another fundamental and dramatically evolving technology is microelectronics, where elements
down to nanometer dimensions are manipulated (Novikov et al., 2007). In this case and, in particular, in very large scale
integration (VLSI), there is the challenge of system integration and manipulation over multiple scales, involving several
orders of magnitude. All known displacement measurement principles, with a resolution of approximately 1 nm or less,
are limited to a very small measurement range (Brand and Herrmann, 1996). Consequently, in nanotechnology,
multiscale exploration remains a major challenge (Chassagne et al., 2010). In a multiscale positioning system, the
measurement of displacement with high accuracy is of most importance (Kuo et al., 2007), and among the several
techniques, optical interferometry are broadly employed, due to simplicity and high sensitivity.
However, despite of being essential in MAPFAs design, compliant mechanisms produce an undesired effect: the
displacement in a direction (generated displacement) results in a spurious displacement (coupled displacement) in the
other direction. Consequently, measurements of displacement of such structures are of great importance to verify
analytical and FEM models, to access the material and device properties, to detect defects and to determine MAPFA
performance. Optical techniques, such as the interferometric one, provide several advantages relative to other
conventional methods for validating, testing and characterizing MAPFAs.
Finally, because the MAPFAs discussed in this work were developed for static or quasi-static practical applications,
far from mechanical resonances, their resonant frequencies must be high enough to guarantee linear operation of the
transducers in a wide range of low frequencies and small time response. As the first resonance frequency usually is not
considered in their numerical design procedures, interferometry plays an important role in the measurement of these
resonance frequencies, bandwidth, linearity, and coupling rate.
There is a general agreement that heterodyne interferometry techniques, with picometer resolution, are useful when
the displacement exceeds an interference fringe, while homodyne techniques are often employed when the displacement
does not exceed the fringe spacing (Smith et al., 2009). However, homodyne interferometers still have interesting
characteristics, such as ease of use, low cost and small periodic error level, which are much more pronounced in
heterodyne interferometers (Fan et al., 2007). Moreover, with the availability of inexpensive processing power, some
computational signal processing complexity is often preferable to recover the optical path length from the measured
photocurrent, instead of additional optics and electronics that are required by other techniques.
In this work, a low cost open-loop homodyne Michelson interferometer is utilized to experimentally detect the
nanovibrations of MAPFAs, based on the spectral analysis of the interferometric signal. Based on the well-known J1/J3
phase demodulation method (Deferrari et al., 1967), a new and improved version is proposed, which presents the
following characteristics: is self-consistent, is immune to fading, does not present phase ambiguity problems and
presents a wide phase demodulation dynamic range. Modulation indexes values up to 130 rad have been measured with
high accuracy, and this limit is determined only by the detector and data acquisition bandwidths employed. This range
is sufficient to analyze the MAPFAs discussed in section 2.
2. MULTI-ACTUATED PIEZOELECTRIC FLEXTENSIONAL ACTUATORS
Topology optimization is a powerful structural optimization technique that combines the FEM with an optimization
algorithm to find the optimal material distribution inside a given domain bounded by supports and applied loads that
must contain the unknown structure (Carbonari et al., 2009). The objective of topology optimization is to determine the
holes and connectivities of the structure by adding and removing material in the extended fixed domain. The main
question to be addressed in topology optimization is how to change the material from zero (void) to one (material).
Since piezoelectric actuators are usually two dimensional (2D) devices, the examples presented herein are limited to
two dimensions (2D plane stress).
Thus, the development of MAPFA mechanisms requires the design of actuated compliant mechanisms that can
perform complex movements. In addition, when multiple piezoceramic devices are involved, coupling effects in their
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movements become critical, especially the appearance of undesired movements, which makes the design task very
complex. One way to avoid undesirable coupling effects is to use a systematic design method, such as topology
optimization, with appropriate formulation of the optimization problem (Carbonari et al., 2005a). Essentially, the
optimization problem is posed as the design of a flexible structure that maximizes different output displacements or
output forces in different specified directions and points of the domain, in response to different excited piezoceramic
portions, with particular attention to minimizing the effects of movement coupling. The MAPFAs considered here
operate in quasi-static or low-frequency applications, that is, lower than their first resonance frequency, and so, the
inertia effects are neglected (Carbonari et al., 2009).
Figure 2 illustrates the various stages involved in the topology optimization procedure to design a MAPFA
(Carbonari et al., 2009). First, an initial design domain is defined [Fig. 2a)], which is limited by the device boundary
conditions such as regions where there are restraints and applied loads. This domain is discretized into finite elements
[Fig. 2b)] and it is an input to the topology optimization software. The objective of topology optimization is to
determine the voids and connectivities of the structure by adding and removing material in the extended fixed domain
[Fig. 2c)]. The finite element model domain is not changed during the optimization process. The software gives as a
result an optimized material distribution in the design domain, which must be interpreted [Fig. 2d)], verified [Fig. 2e)],
and manufactured [Fig. 2f)]. By changing coupling structure topologies, novel designs of flextensional actuators are
obtained for different applications. A detailed review of modeling for displacement amplified piezoelectric actuators,
the FEM using ANSYS, and their application to compliant mechanism design is given in (Carbonari et al., 2005a;
Carbonari et al., 2007, Carbonari et al., 2009).
(a)
(b)
(c)
(d)
(e)
(f)
Figure 1. Topology optimization procedure for designing piezoelectric mechanisms (u is the desired output
displacement) (Carbonari et al., 2009). a) Initial domain, b) Domain discretized, c) Topology obtained,
d) Interpretation, e) Verification, and f) Manufactured.
The piezoelectric multi-actuator used in this work consists in a coupling structure actuated by two piezoceramics
where each piezoceramic is responsible for actuating a specific multi-actuator movement. In addition, there is a
coupling between actuated displacements due to the fact that it is a flexible structure. Thus, when a piezoceramic is
excited to generate a desired displacement, other undesired displacements may also be generated. Figure 2 illustrates the
MAPFA prototype tested in this work, which uses two piezoceramics (PZT-5A 20 mm x 5 mm x 1 mm in directions 1, 2
and 3, respectively) and an aluminum flexible structure manufactured using a wire EDM (Electrical Discharge
Machining) machine. These piezoceramics are polarized in the 3-direction and the electrode surfaces are normal to this
direction. Displacement measurement was performed under mechanically ‘free’ condition. The experiment was
performed in air.
Experimental displacement measurements were performed to evaluate the prototype performances in terms of
movement generated and coupled displacements. Since these devices are used in static or quasi-static mode,
displacements are determined by considering low excitation frequencies. In general, the XY actuator is build to achieve
the maximum displacement in X and Y direction with minimum crosstalk between both directions.
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(b)
(a)
Figure 2. XY nanopositioner prototype. a) Lateral view. b) Top view.
3. OPTICAL INTERFEROMETRY AND CLASSICAL J1/J3 METHOD
In this work, the experiment is based on a bulk homodyne Michelson interferometer (Udd and Spillman, 2011).
Figure 3 shows the schematic of the experimental setup. The laser source is a linearly polarized He-Ne laser. A 50%
beam splitter divides the beam into two equal parts. One (the reference beam) is sent to a fixed mirror; the second (the
measurement beam) is sent to the MAPFA surface. The back reflected beam is superimposed to that coming from the
fixed mirror on the beam-splitter and is sent to the detection assembly.
Figure 3. Homodyne Michelson interferometer used to characterize the MAPFA.
At the output of a homodyne Michelson interferometer, the photodetected voltage can be related to dynamic phase
shift,  (t ) , as follows (Udd and Spillman, 2011):
v (t ) 
I
{1  F cos[0 (t )   (t )]} ,
2
(1)
where I is a proportionality constant that depends on the optical source intensity, the photodetector responsivity and the
signal conditioning circuit gain. Factor F is related to the interference fringe visibility and  0 (t ) is a static phase (in
principle). The phase 0 (t ) should ideally remain constant, but in practice it is susceptible to signal fading, i.e., low
frequency fluctuations in the local temperature of the interferometer, external mechanical vibrations, and air turbulence,
that produce differential drifts on the interferometer arms, causing large phase shifts in 0 (t ) (Sheem et al., 1982).
Consequently, random variations in the amplitude of the photodetected signal occur, severely limiting the direct
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applicability of the sensor system. For this reason, adequate techniques must be applied to obtain a linear relationship
between the output electrical signal and induced phase-shift.
When a piezoactuator is excited with a sinusoidal voltage V (t )  Vpeak sin(s t ) , with amplitude Vpeak and angular
frequency s, the signal phase-shift between the sensor and reference beams in the interferometer can be written as
follows (Udd and Spillman, 2011):
 (t )  x sin(s t ),
(2)
where x is the modulation index or amplitude of the phase change. By detecting the light intensity at the interferometer
output, the electric voltage can be estimated from (Sudarshanam and Srinivasan, 1989):
v(t )  A  B{cos0 (t )[ J 0 ( x)  2 J 2 ( x) cos 2s t  ...]  sin 0 (t )[2 J1 ( x) sin s t  2 J 3 ( x) sin 3s t  ...]},
(3)
where A and B are constants, and the terms J m (x) are Bessel functions of the first kind and order m. Once x is
determined, the piezoactuator vibration amplitude or surface displacement at the measurement point is given by (Udd
and Spillman, 2011):
u   x / 4
(4)
where is the light wavelength. The interferometry problem is posed as the detection of very low values of modulation
index x under high values of random phase shift 0 (t ) and under the presence of electronic noise.
Methods of spectrum analysis of the interferometer output can be used to measure displacements by simple passive
phase-detection schemes, independently of fading (Deferrari et al., 1967; Sheem et al., 1982; Sudarshanam and
Srinivasan, 1989; Udd and Spillman, 2011). Dynamic phase demodulation methods using spectral analysis techniques
are classic tools used since the 1920’s (Thomas and Warren, 1928), that were established and evolved significantly
after the 1960’s, with the advent of the laser. For example, the J1/J3 method was proposed in the classic article of
Deferrari et al. (1967), together with other methods like the J1/J2 and J1max. Nevertheless, some drawbacks or
misunderstandings about some of these methods usually are not pointed out in the literature.
In order to understand the J1/J3 method, consider the even and odd harmonic amplitudes of v(t) in Eq. (3):
V2 n  2 BQJ 2 n ( x) and V2 n1  2 BPJ 2 n1 ( x)
(5)
respectively, and n = 1, 2, 3, …, where P  sin 0 (t ) and Q  cos 0 (t ) . For example, the well known J1/J3 method uses
the fundamental (V1) and third (V3) harmonics of the interferometric signal and calculates the ratio
V1 / V3  J1 ( x) / J 3 ( x) . In this operation only the odd spectral components are involved and, consequently, the amplitude
factors associated to V1 and V3, related to intensity of the optical sources, photodetector responsivity and fringe
visibility, are cancelled and the computation of x does not depend on 0 (t ) . As a result, the J1/J3 method is immune to
fading and does not depend on the values or instabilities in the amplitude factors.
However, there are some troubles with the J1/J3 technique such as the difficulty to solve a transcendental equation
and the phase ambiguity problem. Actually, for a given V1/V3 ratio there are infinite solutions for x, due to the
oscillatory behaviour of the Bessel functions Jn(x). This characteristic restricts the application of the method to those
cases when x increases from zero, in order to track the evolution of the transcendental equation solution. Another
problem refers to the algebraic sign of the Bessel functions (as an example, for x>3.83 rad, J1(x) becomes negative for
the first time), that must be considered when using the magnitude factors. Furthermore, Deferrari et al. (1967) did not
take into account the electronic noise in the photodetection method, which varies the SNR concomitantly with the
variation of 0 (t ) . Then, depending on the value of x, 0 (t ) and order n, the magnitudes of V1 and (principally) V3 can
present the same order of magnitude of noise, or even less. According to Sudarshanam (1992) the MDPS (Minimum
Detectable Phase Shift) is the value of x that satisfies the condition x  x (in analogy to the signal-to-noise ratio,
SNR), where x  x' x is the absolute error between the estimated value x' and the expected value x of the modulation
index, in the presence of noise. For the J1/J3 method and considering 1/f 2 power noise (red noise), the MDPS is 0.18 rad.
Below this value, noise predominates (over the component V3) and the estimated value of x diverges 100% from the
expected value. Consequently, the J1/J3 method does not operate for small values of x.
In this work, a new and improved version of the J1/J3 method is proposed, which overcomes most of the limitations
cited previously.
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4. THE GENERALIZED J1/J3 INTERFEROMETRIC METHOD
In this work, an improved version of the J1/J3 method is proposed, which is here denominated “generalized J1/J3
method”. The objective is to establish a standard and computationally simple procedure to estimate values of x as high
as 130 rad (or higher). The challenge is to do it when there is fading, which is manifested by the low frequency random
phase  0 (t ) . The first step is to determine the magnitude spectrum of the photodetected signal in Eq. (1), as given by
Eq. (3). Figure 4 shows a typical a.c. instantaneous waveform and the corresponding Fourier spectrum presented at the
interferometer output, like that shown in Fig. 3, where V1, V2, V3, etc., are the voltage magnitudes of the fundamental
and the higher order harmonics. The J1/J3 generalized computational algorithm is based on the value of n=fn/f0, where fn
is the frequency corresponding to higher magnitude and f0 is the fundamental frequency of the MAPFA drive voltage
(in Fig. 4, n=9).
Figure 4. Typical output interferometric signal observed when the MAPFA is driven with a sinusoidal voltage.
a) Photodetected signal. b) The corresponding spectrum
The determination of the modulation index x is performed using software Matlab, more specifically, the fzero
command, which tries to find a zero of a certain function f(x) near x0, if x0 is a scalar (Attaway, 2011). However, two
ranges of values of n must be tested: n  3 and n  3 . Basically, if n  3 the algorithm involves two steps:
Step1: Obtain, as a first approximation for x, named as x0, the value of x where the difference:
 J ( x )
V
 n
d ( x)   n
 J n  2 ( x) Vn  2



(6)
becomes minimum within the range between xmin and xmax, where xmin is the first point of intersection between the
functions Jn−2(x) and Jn(x), and xmax is the first point of intersection between Jn(x) and Jn+2(x);
Step 2: Given x0 obtained in Step 1, use the fzero command to find the root of the function f(x’), such as:
 J ( x ' )
Vn
f ( x' )   n

J
(
x
'
)
V
 n 2
n 2



(7)
near the value of x0 found in Step 1, where x’ is the estimated (or measured) modulation index and x is the expected
value of the modulation index.
However, if n  3 , ambiguity cases occur when the harmonic with n = 1 or n = 2 becomes maximum for the second
time when x increases, depending on the value of random phase shift  0 (t ) .
Actually, an important detail resides in the envelope of maxima and the corresponding value of n, as x increases. As
there is interest in the maximum magnitude for each value of x consider, for example, the particular case where
0   / 2 rad. First, there is a region where V1(x) (given by Eq. (5)) is maximum, for 0 ≤ x ≤ 3.0558 rad; then V3(x) is
maximum from 3.0558 to 5.1363 rad; then V1(x) is maximum again, from 5.1363 to 5.6164; then V5(x), for 5.6164 ≤ x ≤
7.5019 rad. The authors have observed that for higher values of x after this point, the sequence of maxima are related to
functions V7(x), V9(x), V11(x), etc. This was verified up to 130 rad, but is possibly valid until infinity. Consequently,
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there is only one ambiguity, related to function V1(x), which is maximum at two disjoint intervals. The challenge is to
realize a similar analysis when there is fading, which is manifested by low frequency and time varying random phase
 0 (t ) . However, it can be shown that same conclusion are obtained for values of  0 (t ) in the range π/2 − 0.3156π ≤
 0 (t ) ≤ π/2 + 0.3156π and 3π/2 − 0.3156π ≤  0 (t ) ≤ 3π/2 + 0.3156π rad.
On the other hand, for the particular case where 0   rad, the even harmonics are higher than the odd ones, and
increasing values of x, the harmonic with order n = 2 is larger, then n = 4, then again n = 2, then n = 6, and so on. The
same conclusion is obtained for values of  0 (t ) in the range −0.1974π ≤  0 (t ) ≤ 0.1974π and π −0.1974π ≤  0 (t ) ≤
π+0.1974π rad.
Finally, a detailed analysis for the remaining values of  0 (t ) reveals that the ambiguity cases (when harmonic with
n = 1 or n = 2 become maximum for the second time) do not occur.
Thus, in the case for n  3 , the authors found out that the ratio Vn 4 / Vn must be calculated. If Vn  4 / Vn  0.6 ,
then this corresponds to the standard case. Consequently, the value of n should be maintained and Steps 1 and 2 should
be applied. Otherwise, if Vn  4 / Vn  0.6 , then the value of n is replaced by (n +2) and Steps 1 and 2 should be applied
to this new value of n. This is the treatment for the ambiguity case. However, it must be adopted xmin=0+ (or xmin=eps in
Matlab) for both, n=1 and n=2; the value of xmax remains the same, i.e., the first intercept between Jn(x) and Jn+2(x).
An important advantage of the method is that there is no problems with singularities of the type  0 (t ) = ℓπ/2 rad for
ℓ = ±1, ±2, ±3, …, and it is not necessary to use any algorithm to detect Bessel function algebraic sign inversion, as
occurs with the modified J1...J4 method (Jin et al., 1991). This is valid for any value of x larger than the MDPS, which
was verified in this work up to 130 rad.
The proof of the J1/J3 algorithm is too long to be presented here and it will be submitted for publishing in a regular
paper.
5. EXPERIMENTAL RESULTS
MAPFA displacements are measured by using a low cost open loop homodyne bulk Michelson interferometer,
where a He-Ne laser source (Newport, 15 mW, λ = 632.8 nm) and a PIN photodiode (ThorLabs, PDA 55) are used. The
output signal was digitized by an oscilloscope (Tektronix TDS2022) and transferred to a computer to be post processed.
A commercially available reflective sheet (3M Scotch-lite 7610) covered with 60 m diameter spherical glass lenses are
mounted on the surface of the sample (3M Scotchlite, 2003). The input sinusoidal voltage amplitude is varied from 0 to
180 V (peak) and the linearity curves of the MAPFA are obtained. Output signal like that shown in Fig. 4 are obtained,
and the corresponding spectra are determined. The modulation index x is obtained from the generalized J1/J3 method,
and displacements are calculated by applying Eq. (4). Displacements are determined by considering low excitation
frequencies.
The results for generated displacements at the frequencies of 370 Hz, 930 Hz and 2.811 kHz are presented in Fig. 5.
As can be seen, the generalized J1/J3 method reaches 130 rad (650 nm). There is a linear response starting from 0.18 rad,
and the PFA linear length-to-voltage sensitivities resulted in 0.2684, 0.4102 and 0.8613 nm/V (peak), at the frequencies
of 370 Hz, 930 Hz and 2.811 kHz, respectively.
Figure 5. XY nanopositioner generated movement linearity curves at the frequencies of 370 Hz, 930 Hz and 2,811 Hz.
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In Fig. 6 the results for coupled displacements at the frequencies of 290 Hz, 910 Hz and 1.25 kHz are presented. The
displacement amplitudes in Fig. 5 are smaller than those shown in Fig. 4 for the corresponding voltages. There is a
linear response starting from 0.18 rad, and the MAPFA linear length-to-voltage sensitivities resulted in 0.05806,
0.21924 and 1.1003 nm/V (peak), at the frequencies of 290 Hz, 910 Hz and 1.25 kHz, respectively.
Figure 6. XY nanopositioner coupled movement linearity curves at the frequencies of 290 Hz, 910 Hz and 1,250 Hz.
The frequency response shows the MAPFA mechanical resonance frequencies, as well as the quasi-static range of
operation. The frequency response, in terms of the modulation index- and length-to-voltage ratios, is presented in Fig. 7
(in terms of mean and standard deviation), for frequencies between 76 Hz and 4 kHz. There are mechanical resonances
at the frequencies of 956 Hz, 1.18 kHz, 1.24 kHz, 1.39 Hz and 2.96 kHz in the studied range. At 370 Hz, 930 Hz and
2.811 kHz there was agreement with the linearity responses (slope, in terms of modulation index- and length-to-voltage
ratios) from Fig. 5 and Fig. 6. The PFA –3 dB operation bandwidth is approximately 895 Hz.
Figure 7. Generated XY nanopositioner displacement frequency response, in terms of modulation indexand length-to-voltage ratios.
The MAPFA coupled movement frequency response is shown in Fig. 8, for frequencies between 90 Hz and 1.27
kHz. There are resonances around 370 Hz, 990 Hz, 1,170 Hz, and 1,250 Hz in the studied range. It is observed that near
these frequencies the coupled displacement have similar amplitudes as the generated movements. This means that this
XY nanopositioner would not be useful when operating at resonances.
A quantitative measure of the crosstalk levels between X and Y displacements can be given in terms of percentage
by the following expression (Carbonari et al., 2005b):
S xy  y s / x
(8)
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where x is the desired displacement in the X direction and ys is the spurious displacement found in Y direction when
attempt to displace in X. From the data of generated and coupled movements (Fig. 7 and Fig. 8), the coupling rate Sxy
defined in Eq. (8) can be measured as a function of frequency. The result is shown in Fig. 9.
Figure 8. Coupled XY nanopositioner displacement frequency response, in terms of modulation indexand length-to-voltage ratio.
Coupling rate (dimensionaless)
2
1.5
1
0.5
0
0
200
400
600
800
Frequency [Hz]
1000
1200
1400
Figure 9. Coupling rate Sxy as a function of frequency.
6. CONCLUSIONS
Modulation indexes values up to 130 rad can be measured with high accuracy by using the new generalized J1/J3
method. This range is sufficient to analyze the MAPFAs discussed in section 2. However, it is important to emphasize
that this limit can increase indefinitely, being limited only by the detector and data acquisition frequency bandwidths
employed.
In Precision Engineering, the precision degree is related to the ratio between the resolution/uncertainty range,
defined as precision ratio (PR) by Fan et al. (2007). As it was shown, the generalized J1/J3 method proposed here allows
to obtain one of the best PR values using homodyne interferometers: PR equal 1.1 × 10-3 (range between 10 nm and 8.7
μm). Better values are obtained than those obtained by most successful homodyne systems such as, for example, Jin et
al. (1992), with PR = 3.8 × 10-3, and Ivaschescu (2000), with PR = 6.3 × 10-3. On the other hand, it presents a PR worse
than that obtained by (Heinzel et al., 2010), applied to the detection of gravitational waves: PR = 10-5. In the case of that
article, however, the interest has been focused on expanding the dynamic range by improving resolution, operating from
20 pm to 1 μm, approximately.
Finally, it is noteworthy that the range presented in this work (from 10 nm to 8.7 μm), although it cannot be
expanded to improve the resolution, in principle, can be easily expanded up to unlimited amounts, since is provided
368
with a detection system, and data acquisition bandwidth higher than used herein. So, it is important to clarify that the
not so high value of PR obtained here is due to technological limitations of the experimental setup.
The coupling factor was calculated, showing its frequency dependence, and concluding that, when the XY
nanopositioner is in resonance, the whole structure vibrates, and the generated and coupled displacements are of the
same order of magnitude. Under this condition the MAPFA operation is impracticable.
7. ACKNOWLEDGEMENTS
The authors are grateful to the research agencies FAPESP (project number 2012/01629-7), CNPq (project number
478817/2012-6) and CAPES in Brazil.
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9. RESPONSIBILITY NOTICE
The authors are the only responsible for the printed material included in this paper.
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sinusoidal vibration measurements in a piezoelectric xy

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